Finsler geometries on strictly accretive matrices

Abstract

In this work we study the set of strictly accretive matrices, that is, the set of matrices with positive definite Hermitian part, and show that the set can be interpreted as a smooth manifold. Using the recently proposed symmetric polar decomposition for sectorial matrices, we show that this manifold is diffeomorphic to a direct product of the manifold of (Hermitian) positive definite matrices and the manifold of strictly accretive unitary matrices. Utilizing this decomposition, we introduce a family of Finsler metrics on the manifold and charaterize their geodesics and geodesic distance. Finally, we apply the geodesic distance to a matrix approximation problem, and also give some comments on the relation between the introduced geometry and the geometric mean for strictly accretive matrices as defined by S. Drury in [S. Drury, Linear Multilinear Algebra. 2015 63(2):296-301].